# Asymptotic Eigenfunctions for a class of Difference Operators

## TL;DR

This paper develops asymptotic expansions for eigenfunctions of a class of difference operators with one-well potentials, connecting discrete lattice eigenfunctions to continuous approximations as the lattice spacing becomes small.

## Contribution

It introduces a method to construct formal WKB-type asymptotic expansions for low-lying eigenfunctions of difference operators, bridging discrete and continuous spectral analysis.

## Key findings

- Constructed formal asymptotic expansions for eigenfunctions.
- Linked lattice eigenfunctions to continuous operator eigenfunctions.
- Provided a framework for analyzing low-energy eigenstates in discrete systems.

## Abstract

We analyze a general class of difference operators $H_\varepsilon = T_\varepsilon + V_\varepsilon$ on $\ell^2(\varepsilon \mathbb{Z}^d)$, where $V_\varepsilon$ is a one-well potential and $\varepsilon$ is a small parameter. We construct formal asymptotic expansions of WKB-type for eigenfunctions associated with the low lying eigenvalues of $H_\varepsilon$. These are obtained from eigenfunctions or quasimodes for the operator $H_\varepsilon$, acting on $L^2(\mathbb{R}^d)$, via restriction to the lattice $\varepsilon\mathbb{Z}^d$.